692 CHAPTER 10 Analytic Geometry ‘Are You Prepared?’ Answers 1. x x y y 2 1 2 2 1 2 ( ) ( ) − + − 2. 4 3. 7, 1 { } − − 4. 2, 5 ( ) − − 5. 3; up 6. x 3, 5; 3 ( ) − = 80. Show that an equation of the form Ax Ey A E 0 0, 0 2 + = ≠ ≠ is the equation of a parabola with vertex at 0, 0 ( ) and axis of symmetry the y -axis. Find its focus and directrix. 81. Show that an equation of the form Cy Dx C D 0 0, 0 2 + = ≠ ≠ is the equation of a parabola with vertex at 0, 0 ( ) and axis of symmetry the x -axis. Find its focus and directrix. 82. Challenge Problem Show that the graph of an equation of the form Ax Dx Ey F A 0 0 2 + + + = ≠ (a) Is a parabola if E 0. ≠ (b) Is a vertical line if E 0 = and D AF 4 0. 2 − = (c) Is two vertical lines if E 0 = and D AF 4 0. 2 − > (d) Contains no points if E 0 = and D AF 4 0. 2 − < 83. Challenge Problem Show that the graph of an equation of the form Cy Dx Ey F C 0 0 2 + + + = ≠ (a) Is a parabola if D 0. ≠ (b) Is a horizontal line if D 0 = and E CF 4 0. 2 − = (c) Is two horizontal lines if D 0 = and E CF 4 0. 2 − > (d) Contains no points if D 0 = and E CF 4 0. 2 − < 84. Challenge Problem Let A be either endpoint of the latus rectum of the parabola y y x 2 8 1 0, 2 − − + = and let V be the vertex. Find the exact distance from A to V.† Retain Your Knowledge Problems 85–93 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for subsequent sections, a final exam, or later courses such as calculus. 85. For x y9 36, 2 = − list the intercepts and test for symmetry. 86. Solve: 4 8 x x 1 1 = + − 87. Given tan 5 8 , 2 , θ π θ π = − < < find the exact value of each of the remaining trigonometric functions. 88. Find the exact value: tan cos 3 7 1( ) ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ − 89. Find the exact distance between the points 3, 1 2 ( ) − and 2 3 , 5. ( ) − 90. Find the standard form of the equation of a circle with radius 6 and center 12, 7 . ( ) − 91. Given f x x ln 3 , ( ) ( ) = + find the average rate of change of f from 1 to 5. 92. Express 1 cos34 2 + ° as a single trigonometric function. 93. Solve: x x5 2 4 2 − − = † Courtesy of the Joliet Junior College Mathematics Department DEFINITION Ellipse An ellipse is the collection of all points in a plane the sum of whose distances from two fixed points, called the foci , is a constant. 10.3 The Ellipse Now Work the ‘Are You Prepared?’ problems on page 701. • Distance Formula (Section 1.2, p. 14) • Completing the Square (Section A.3, p. A29) • Intercepts ( Section 1.3 , pp. 20 – 21 ) • Symmetry (Section 1.3, pp. 21–23) • Circles (Section 1.6, pp. 48–52) • Graphing Techniques: Transformations (Section 2.5, pp. 112–120) PREPARING FOR THIS SECTION Before getting started, review the following: OBJECTIVES 1 Analyze Ellipses with Center at the Origin (p. 693) 2 Analyze Ellipses with Center at h, k ( ) (p. 697) 3 Solve Applied Problems Involving Ellipses (p. 700)
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